slug: oeis-a231548-antisigma-decrease-refutation bibkey: krizek2013a231548 doi: null url: https://oeis.org/A231548 triage: theorem motivation_gids:

• D5/S0/Certificates/KrizekAntisigmaDecreaseRefutation

Refutation of the A231548 antisigma gap-three conjecture

Problem

OEIS A231548, NAME (verbatim):

Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2).

COMMENT defining antisigma (verbatim):

Also numbers n such that antisigma(n) < antisigma(n-2), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.

COMMENT conjecture line (verbatim; Jaroslav Krizek, Nov 12 2013):

Conjecture: there are no numbers n such that antisigma(n) < antisigma(n-3). - Jaroslav Krizek, Nov 12 2013

OEIS A024816, NAME (verbatim):

Antisigma(n): Sum of the numbers less than n that do not divide n.

The literal object is antisigma(n) = Σ of d ∈ [1,n] with d ∤ n. Since n always divides itself, this agrees with the A024816 interval ending before n. The conjecture is the claim ∀ n ≥ 3, antisigma(n-3) ≤ antisigma(n).

Motivation

This first-tier OEIS conjecture was printed in 2013 and remains a universal claim about the behavior of the antisigma function. A certified counterexample resolves the literal conjecture without proposing a corrected statement.

Gap

Preregistration issue #7485 and its probe report record searches dated September 13, 2026. OEIS history revisions #1 through #13 only introduce and reword the conjecture: revision #2 says “not numbers”, revision #10 changes that phrase to “no numbers”, and revision #13 retains the current wording. No proof or refutation appears in those revisions. Exact searches returned 0 results on OpenAlex and 0 on MathOverflow. GitHub results were mirrors: the exact conjecture sentence returned 0 results, while the four results for A231548 antisigma reproduced sequence data or the OEIS text. arXiv and Semantic Scholar returned HTTP 429 on both attempts and are ASSUMED-UNVERIFIED. No resolution was found in the bounded surfaces that could be checked. This does not assert exhaustive literature coverage or publication priority.

Route

For every natural n, splitting the interval from one through n into divisors and non-divisors gives antisigma(n) = n(n+1)/2 - sigma(n).

The factorization 332640 = 2⁵·3³·5·7·11 and multiplicativity of the divisor-sum function give sigma(332640) = 1451520. Also 332637 = 3 * 110879, where 110879 is prime, so sigma(332637) = 443520. Therefore antisigma(332640) = 55323399600, while antisigma(332637) = 55323409683. Thus antisigma(332640) < antisigma(332637), contradicting the conjecture at n = 332640.

Falsifier

A proof that antisigma(n-3) <= antisigma(n) for all natural n >= 3 would falsify this refutation. The kernel-checked instance at n = 332640 proves the strict reverse inequality, so such a proof would contradict the formal result.

Evidence

• Lean module: D5/S0/Certificates/KrizekAntisigmaDecreaseRefutation.lean.
• Main theorem: result : ¬ claim, with exactly the std3 axioms propext, Classical.choice, and Quot.sound.
• The recorded kernel type-checking time is 30 ms.
• The orchestrator’s independent sieve over 3 <= n <= 400000 found violations exactly at n = 332640 and n = 360360.
• The probe independently checked the same range and found the same two values, the factorizations of 332640 and 332637, and the primality of 110879.

The bounded sieve is supporting evidence only. The formal result uses the single explicit instance n = 332640 and makes no assertion about any other input.

Triage

theorem. The certified instance at n = 332640 refutes the universal gap-three conjecture. It asserts no corrected monotonicity statement and no claim about other values of n.

ASSUMED-UNVERIFIED

The arXiv and Semantic Scholar searches each returned HTTP 429 twice, so those surfaces were not verified. The literature search is bounded and does not establish exhaustive coverage or publication priority. The statement that 332640 is the least violation is supported only within the checked range 3 <= n <= 400000; no unbounded leastness claim is made.